Find the derivative of the following w.r.t. x :
`tan^(-1)sqrtx`

Answer

To find the derivative of \( y = \tan^{-1}(\sqrt{x}) \) with respect to \( x \), we will use the chain rule and the derivative of the inverse tangent function.

### Step 1: Identify the function and its components
Let \( y = \tan^{-1}(\sqrt{x}) \).

### Step 2: Differentiate using the chain rule
The derivative of \( \tan^{-1}(u) \) with respect to \( u \) is given by:
\[ \frac{d}{du}(\tan^{-1}(u)) = \frac{1}{1 + u^2} \]
In our case, \( u = \sqrt{x} \).

### Step 3: Differentiate \( u = \sqrt{x} \)
Now, we need to find the derivative of \( u \) with respect to \( x \):
\[ \frac{du}{dx} = \frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}} \]

### Step 4: Apply the chain rule
Now we can apply the chain rule:
\[ \frac{dy}{dx} = \frac{d}{du}(\tan^{-1}(u)) \cdot \frac{du}{dx} = \frac{1}{1 + (\sqrt{x})^2} \cdot \frac{1}{2\sqrt{x}} \]

### Step 5: Simplify the expression
Now, simplify the expression:
\[ \frac{dy}{dx} = \frac{1}{1 + x} \cdot \frac{1}{2\sqrt{x}} = \frac{1}{2\sqrt{x}(1 + x)} \]

### Final Answer
Thus, the derivative of \( y = \tan^{-1}(\sqrt{x}) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{x}(1 + x)} \]

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